Generalized Derivations on Prime Near-Rings and Commutativity Conditions

In this paper, we present a systematic investigation of derivations and generalized derivations on prime near-rings, focusing on their influence on the underlying algebraic structure. We examine several annihilation conditions involving derivations and generalized derivations and show that, in a prime near-ring, any element that annihilates the image of a non-zero derivation or its associated generalized derivation must be zero. This result establishes a strong rigidity phenomenon for derivations in prime near-rings. Furthermore, we analyze the interaction between generalized derivations and commutators and prove that certain vanishing or annihilation conditions involving commutators force the near-ring to be commutative. These findings demonstrate that derivations and generalized derivations function not only as algebraic operators but also as effective tools for detecting and controlling non-commutative behavior in near-rings. As a consequence, our results extend several classical commutativity theorems from ring theory to the broader framework of near-rings, thereby contributing to the structural theory of prime near-rings.

Prime near-rings, derivations, generalized derivations, commutativity, annihilators

1. Miyan, P., Demie, S., Markos, A., & Hailu, L. (2024). Some algebraic identities on prime near rings with generalized derivations. Italian J. Pure Appl. Math, 51, 386-397. [Google Scholar] [Crossref]

2. Hasnain, M. M. (2020). Algebraic identities and commutativity in 3-prime nearrings. Palestine Journal of Mathematics, 9(1). [Google Scholar] [Crossref]

3. Bell, H. E., & Daif, M. N. (1995). On derivations and commutativity in prime rings. Acta Mathematica Hungarica, 66(4), 337-343. [Google Scholar] [Crossref]

4. Chen, H. Y. (2013). Generalized derivations centralizing polynomials. Communications in Algebra, 41(7), 2783-2798. [Google Scholar] [Crossref]

5. Brešar, M. (1993). Centralizing mappings and derivations in prime rings. J. algebra, 156(2), 385- 394. [Google Scholar] [Crossref]

6. Posner, E. C. (1957). Derivations in prime rings. Proceedings of the American Mathematical Society, 8(6), 1093-1100. [Google Scholar] [Crossref]

7. Ali, S., Ashraf, M., De Filippis, V., Oukhtite, L., & Rehman, N. U. (2025). Differential Identities in Rings and Algebras and their Applications. CRC Press. [Google Scholar] [Crossref]

8. Wubshet, T., & Abebaw, T. (2019). Some results on the commutativity of prime near-rings using generalized derivations. SINET: Ethiopian Journal of Science, 42(1), 10-17. [Google Scholar] [Crossref]

9. Yusuf, T. A., Madugu, A., & Babangida, B. (2017). Some proofs on Commutativity Conditions in Prime Near-Rings. [Google Scholar] [Crossref]

10. Rumah, H. M., Balogun, F., & Yusuf, T. A. (2023). Commutativity of prime rings with multiplicative (generalized-reversed) derivation. Science World Journal, 18(3), 386-388. [Google Scholar] [Crossref]

11. Golbasi, O., & Oguz, S. (2012). Notes on (σ, τ)-derivations of lie ideals in prime rings. Communications of the Korean Mathematical Society, 27(3), 441-448. [Google Scholar] [Crossref]

12. Madugu, A., & Yusuf, T. A. (2025). A Work on Generalized Reverse Derivation and Skew- Derivation on Prime Near-Rings. International Journal of Research and Innovation in Social Science, 9(4), 5227-5232. [Google Scholar] [Crossref]

13. Tasiu Abdullahi Yusuf and Abubakar Salisu (2025). Exploring Prime Near-Rings and Semiprime Rings Under Skew and Generalized Skew Derivations, International Journal of Latest Technology in Engineering, Management & Applied Science, 14, issue 9, p. 347-352. [Google Scholar] [Crossref]

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